Which of the following is not a property of the $F$ distribution?

• A. The shape of the $F$ distribution depends on two different degrees of freedom.
• B. The $F$ distribution is always positive and cannot take negative values.
• C. The mean of the $F$ distribution exists only if the denominator degrees of freedom are greater than $2$.
• D. The $F$ distribution is symmetric about its mean.

Answer: (D)

Verified step by step guidance

1. Recall the definition of the F distribution: it is the distribution of the ratio of two independent chi-square variables each divided by their respective degrees of freedom.
2. Understand that the shape of the F distribution depends on two parameters: the numerator degrees of freedom (df1) and the denominator degrees of freedom (df2). This means the shape changes as these degrees of freedom change.
3. Recognize that the F distribution is defined only for positive values because it is a ratio of variances, which are always non-negative. Therefore, it cannot take negative values.
4. Know that the mean of the F distribution exists only if the denominator degrees of freedom (df2) is greater than 2. This is a key property related to the moments of the distribution.
5. Finally, consider the symmetry of the F distribution: unlike the normal distribution, the F distribution is not symmetric about its mean; it is right-skewed. Hence, the statement that it is symmetric about its mean is incorrect.